Cosmology, New Entropy and Thermodynamics of Apparent Horizon

1. Introduction

It is know that black holes can be described by thermodynamics with entropy being proportional to the horizon area [1,2] and temperature is linked with the surface gravity. Thus, gravity is related to ordinary thermodynamics [3,4,5,6]. From the first law of apparent horizon thermodynamics Friedmann’s equations also can be derived [7,8,9,10,11,12,13,14,15,16,17]. Different entropies were studied in [18,19,20,21,22,23,24] which can lead to modified Friedmann’s equations. Other holographic dark energy models were considered in Refs. [25,26,28,29,30,31,32]. Holographic energy densities, depending on the form of entropy, may describe the dark energy which drives the universe to accelerate [33,34]. The nature of dark energy is unknown and can be described by the LCDM model. We propose here new apparent horizon entropy $S_K=S_{BH}/(1+\gamma S_{BH}^2)$ with $S_{BH}$ being the Bekenstein–Hawking (BH) entropy, that lead to the presence of dark energy so that our model is alternative to the LCDM model. The $S_K$ entropy becomes zero when the BH entropy vanishes and is the monotonically increasing function of the BH entropy $S_{BH}$ and is positive. When parameter $\gamma$ vanishes we arrive at the BH entropy. It should be noted that the apparent horizon thermodynamics leads to the Friedmann equations, in the framework of Einstein’s gravity, only when the matter is a perfect fluid with equation of state (EoS) $p=-\rho$ with p being the matter pressure and $\rho$ is the density energy of matter [16]. Modifying the BH entropy $S_{BH}$ by apparent horizon entropy $S_K$ we study the general case of EoS for barotropic perfect fluid $w=p/\rho$. It is worth noting that the long-range gravitational interactions are described by generalized entropies. It will be shown that $S_K$ entropy leads to modified Friedmann’s equations that describe the universe inflation. In our approach the cosmological constant is dynamical and it explains the presents of dark energy.

2. New Entropy

Let us consider new entropy

$$S_K=-\sum _{i=1}^W{\displaystyle \frac{p_{i}ln{p}_i}{1+\gamma {(ln\left(p_i\right))}^2}},$$

(1)

with W being a number of states and each state has a probability $p_i$ with the probability, $\gamma$ is a free parameter. The summation in Equation (1) is performed over all possible system microstates. In the case when $\gamma =0$ entropy (1) becomes the Gibbs entropy

$$S_G=-\sum _{i=1}^W{p}_{i}ln\left(p_i\right).$$

(2)

If each microstate is populated with equal probability, $1/p_i=W$ ($i=1,2,...,W$), then Equation (2) is converted into the Boltzmann entropy $S_B=ln\left(W\right)$. By virtue of $1/p_i=W$, we find from Equation (1)

$$S_K={\displaystyle \frac{ln\left(W\right)}{1+\gamma {(ln\left(W\right))}^2}}.$$

(3)

The BH entropy is $S_{BH}=ln\left(W\right)$ and from Equation (3) one obtains

$$S_K={\displaystyle \frac{S_{BH}}{1+\gamma S_{BH}^2}}.$$

(4)

From Equation (4) we find at $\gamma =0$ the BH entropy $S_{BH}$. When A and B are two probabilistically independent systems, one has $p_{ij}^{A+B}=p_i^A{p}_j^B$ and entropy $S_K$ being the nonadditive entropy because $S_K(A+B)\ne S_K\left(A\right)+S_K\left(B\right)$.

3. Thermodynamics of Apparent Horizon

We consider here the FLRW flat universe with the metric

$$d{s}^2=-d{t}^2+a{\left(t\right)}^2(d{r}^2+r^{2}d{\Omega }_2^2).$$

(5)

In Equation (5) $a\left(t\right)$ is a scale factor and $d{\Omega }_2^2$ represents the line element of 2-dimensional unit sphere. In the FLRW universe the radius of the apparent horizon $R_h=a\left(t\right)r$ is given by

$$R_h={\displaystyle \frac{1}{H}},$$

(6)

where $H=\dot{a}\left(t\right)/a\left(t\right)$ is the Hubble parameter of the universe, and dot over the scale factor being the derivative with respect to the cosmological time t. Inside the space, the total energy is defined as

$$E=\rho V_h={\displaystyle \frac{4\pi }{3}}\rho R_h^3,$$

(7)

where, $\rho$ is the energy density of matter fields and the first law of apparent horizon thermodynamics reads

$$dE=-T_{h}d{S}_h+Wd{V}_h.$$

(8)

In cosmology the work density W is given by

$$W=-{\displaystyle \frac{1}{2}}\mathrm{Tr}\left(T^{\mu \nu }\right)={\displaystyle \frac{1}{2}}(\rho -p),$$

and p is the matter pressure. The apparent horizon temperature is represented as

$$T_h={\displaystyle \frac{H}{2\pi }}\left|1+{\displaystyle \frac{\dot{H}}{2H^2}}\right|.$$

(9)

Making us of Equations (6), (7), and (9), we obtain from first law of apparent horizon thermodynamics (8) the equation as follows:

$${\displaystyle \frac{H}{2\pi }}\left|1+{\displaystyle \frac{\dot{H}}{2H^2}}\right|d{S}_h=-{\displaystyle \frac{4\pi }{3H^3}}d\rho +{\displaystyle \frac{2\pi (\rho +p)}{H^4}}dH.$$

(10)

With the aid of the energy momentum conservation (the continuity equation)

$$\dot{\rho }=-3H(\rho +p).$$

(11)

we find Equation (10) in the form

$${\displaystyle \frac{H}{2\pi }}\left|1+{\displaystyle \frac{\dot{H}}{2H^2}}\right|{\dot{S}}_h=-{\displaystyle \frac{4\pi \dot{\rho }}{3H^3}}\left(1+{\displaystyle \frac{\dot{H}}{2H^2}}\right).$$

(12)

4. Modified FLRW Equations

Assuming that $\dot{H}\ge 2H^2$ and making use of Equations (11) and (12), we find

$${\displaystyle \frac{H}{2\pi }}{\dot{S}}_h={\displaystyle \frac{4\pi (\rho +p)}{H^2}}.$$

(13)

By virtue of our entropy (4) ($S_h=S_K$) and BH entropy

$$S_{BH}={\displaystyle \frac{\pi }{G{H}^2}}$$

(14)

and utilizing Equations (13) and (14), one obtains the modified Friedmann equation

$${\displaystyle \frac{\dot{H}(1-\gamma {\pi }^2/{\left(G{H}^2\right)}^2)}{{(1+\gamma {\pi }^2/{\left(G{H}^2\right)}^2)}^2}}=-4\pi G(\rho +p).$$

(15)

Equation (15), as $\gamma \to 0$, is converted into the usual Friedmann equation within general relativity for flat universe. Integrating Equation (15) and making use of Equation (11) we find the second modified Friedmann equation

$$H^2+{\displaystyle \frac{b{H}^2}{b+H^4}}-2\sqrt{b}arctan\left({\displaystyle \frac{H^2}{\sqrt{b}}}\right)={\displaystyle \frac{8\pi G}{3}}\rho ,$$

(16)

where we have defined parameter $b={\pi }^2\gamma /G^2$. When $b=0$ ($\gamma =0$) Equation (16) becomes the FLRW equation for flat universe in the framework Einstein’s gravity. Introducing the effective (a dynamical) cosmological constant

$${\Lambda }_{eff}=6\sqrt{b}arctan\left({\displaystyle \frac{H^2}{\sqrt{b}}}\right)-{\displaystyle \frac{3b{H}^2}{b+H^4}},$$

(17)

Equation (16) can be put into the usual form of Friedmann’s equation

$$H^2={\displaystyle \frac{8\pi G}{3}}\rho +{\displaystyle \frac{{\Lambda }_{eff}}{3}}.$$

(18)

The dynamical cosmological constant ${\Lambda }_{eff}$ versus H at some parameters $b={\pi }^2\gamma /G^2$ is depicted in Figure 1.

In accordance with Figure 1, ${\Lambda }_{eff}$ increases as the Habble parameter H increases. As $H\to \infty$ the dynamical cosmological constant becomes ${\Lambda }_{eff}\to 3\pi \sqrt{b}$. At fixed H, when b increases the dynamical cosmological constant ${\Lambda }_{eff}$ also increases. Making use of Equations (17) and (18) we obtain the dark energy density

$${\rho }_D={\displaystyle \frac{3}{8\pi G}}\left[2\sqrt{b}arctan\left({\displaystyle \frac{H^2}{\sqrt{b}}}\right)-{\displaystyle \frac{b{H}^2}{b+H^4}}\right].$$

(19)

Let us define the normalized density parameters ${\Omega }_m=\rho /\left(3M_P^2{H}^2\right)$ and ${\Omega }_D={\rho }_D/\left(3M_P^2{H}^2\right)$, where $M_P=1/\sqrt{8\pi G}$ is the reduced Plank mass. Then from Equations (17), (18) and (19), one finds the equation ${\Omega }_m+{\Omega }_D=1$. By virtue of Equations (17),(18) and (19) we obtain the normalized density of the matter ($w=0$)

$${\Omega }_m=1-{\displaystyle \frac{2\sqrt{b}}{H^2}}arctan\left({\displaystyle \frac{H^2}{\sqrt{b}}}\right)+{\displaystyle \frac{b}{b+H^4}}.$$

(20)

The ${\Omega }_m$ versus H is plotted in Figure 2.

For the current era ${\Omega }_m\approx 0.26$, and in accordance with Figure 2, one can obtain the corresponding parameters b and H. Assuming that dark substance obeys ordinary conservation law, and there is no mutual interaction between the cosmos components, we find from the continuity equation the dark energy pressure

$$p_D=-{\displaystyle \frac{{\dot{\rho }}_D}{3H}}-{\rho }_D.$$

(21)

With the help of Equations (19) and (21) one obtains the pressure

$$p_D=-{\displaystyle \frac{b(b+3H^4)\dot{H}}{4\pi G{(H^4+b)}^2}}-{\displaystyle \frac{3}{8\pi G}}\left[2\sqrt{b}arctan\left({\displaystyle \frac{H^2}{\sqrt{b}}}\right)-{\displaystyle \frac{b{H}^2}{b+H^4}}\right].$$

(22)

Making use of Equations (15), (19) and (22) one finds EoS for the dark energy $w_D=p_D/{\rho }_D$,

$$w_D={\displaystyle \frac{b(b+3H^4)(1+w)}{H^4(H^4-b)}}\left({\displaystyle \frac{H^2}{2\sqrt{b}arctan(H^2/\sqrt{b})-b{H}^2/(H^4+b)}}-1\right)-1.$$

(23)

The $w_D$ versus H is plotted in Figure 3.

In accordance with left panel of Figure 3 at $b=1$ and $w=0,-1/3,-2/3$ when w increases, EoS parameter for dark energy $w_D$ also increases (at $H>1.5$). According to right panel of Figure 3 at $w=1$ and $b=2,3,4$ when b increases, $w_D$ also increases (at $H>1.5$). From Equation (23), it follows that ${lim}_{H\to \infty }{w}_D=-1$ so that the dynamical cosmological constant leads to EoS of dark energy $w_D=-1$ at large Habble parameter H (small apparent horizon radius $R_h$). As a result, universe inflation is due to dynamical cosmological constant. When $H\to 0$ ($R_h\to \infty$) the dynamical cosmological constant vanishes (${\Lambda }_{eff}\to 0$). Thus, after Big Bang ($R_h\approx 0$) we have the de Sitter space, $p_D+{\rho }_D=0$.

According to the second law of apparent horizon thermodynamics we have the requirement ${\dot{S}}_K\ge 0$ and from Equation (4) one obtains $(1-\gamma S_{BH}^2){\dot{S}}_{BH}/{(1+\gamma S_{BH}^2)}^2\ge 0$ or $(1-\gamma S_{BH}^2)\ge 0$ and ${\dot{S}}_{BH}=-2\pi \dot{H}/\left(G{H}^3\right)\ge 0$. As a result, these requirements, for positive Hubble parameter, lead to $\dot{H}\le 0$ and $(1-\gamma S_{BH}^2)\ge 0$. Then from Equation (15) we find that at $1-\gamma {\pi }^2/{\left(G{H}^2\right)}^2)\ge 0$ one has $\rho +p\ge 0$ and for the positive energy density we obtain for EoS parameter the requirement $w\ge -1$.

Now we explore the redshift $z=a_0/a\left(t\right)-1$, where $a_0$ corresponds to a scale factor at the current time. From the continuity Equation (11) and EoS $p=w\rho$ we obtain the density energy of matter

$$\rho ={\rho }_0{\left({\displaystyle \frac{1+z}{a_0}}\right)}^{3(1+w)},$$

(24)

where ${\rho }_0$ is the density energy of matter at the present time. With the aid of Equations (16) and (24) one finds equation as follows:

$$H^2+{\displaystyle \frac{b{H}^2}{b+H^4}}-2\sqrt{b}arctan\left({\displaystyle \frac{H^2}{\sqrt{b}}}\right)={\displaystyle \frac{8\pi G}{3}}{\rho }_0{\left({\displaystyle \frac{1+z}{a_0}}\right)}^{3(1+w)}.$$

(25)

making use of Equations (24) and (25) we obtain the redshift

$$z=a_0{\left({\displaystyle \frac{3}{8\pi {\rho }_{0}G}}\left(H^2+{\displaystyle \frac{b{H}^2}{H^4+b}}-2\sqrt{b}arctan\left({\displaystyle \frac{H^2}{\sqrt{b}}}\right)\right)\right)}^{1/\left(3\right(1+w\left)\right)}-1.$$

(26)

We plotted the function of Habble parameter versus redshift z in Figure 4 for $G={\rho }_0=a_0=1$.

Figure 4 shows that sas redshift z increases the Habble parameter H also increases. According to left panel of Figure 4, when EoS parameter w increases, at fixed z, the H also increases. Right panel of Figure 4 shows that when parameter b increases at fixed z the Habble parameter also increases.

Let us investigate the phases of acceleration and deceleration of the universe. The deceleration parameter is given by

$$q=-{\displaystyle \frac{\ddot{a}a}{{\dot{a}}^2}}=-1-{\displaystyle \frac{\dot{H}}{H^2}}.$$

(27)

If $q<0$ we have the acceleration phase but when $q>0$ the phase of the universe deceleration takes place. By virtue of Equations (15), (24) and (27) we obtain the deceleration parameter as a function of redshift z

$$q={\displaystyle \frac{4\pi G{\rho }_0(1+w){(H^4+b)}^2}{H^6(H^4-b)}}{\left({\displaystyle \frac{1+z}{a_0}}\right)}^{3(1+w)}-1.$$

(28)

With the help of Equations (25) and (28) one finds the deceleration parameter q in the form

$$q={\displaystyle \frac{3(1+w){(H^4+b)}^2}{2H^6(H^4-b)}}\left(H^2+{\displaystyle \frac{b{H}^2}{b+H^4}}-2\sqrt{b}arctan\left({\displaystyle \frac{H^2}{\sqrt{b}}}\right)\right)-1.$$

(29)

In Figure 5 we depicted the deceleration parameter q versus the Hubble parameter H.

For some values of w and $\gamma$ there are two phases: inflation ($q<0$) and deceleration ($q>0$) but at some w and $\gamma$ we have only eternal university acceleration (inflation), $q<0$. In accordance with figure, when redshift z increases the deceleration parameter q also increases. According to left panel of Figure 5, when EoS parameter w increases the deceleration parameter q also increases at fixed $\gamma$. At $w=-0.1$ and $-0.3$ there are two phases, acceleration $q<0$ and deceleration $q>0$ but at $w=-0.5$ we have the acceleration phase (the eternal inflation). Accordance to right panel of Figure 5, when parameter b (and $\gamma$) increases, at fixed w, the deceleration parameter q decreases. At $b=4$ and $b=6$ we have two phases: acceleration and deceleration.

Making use of Equation (29) one obtains the asymptotic

$$\underset{H\to \infty }{lim}q={\displaystyle \frac{3w+1}{2}}.$$

(30)

Equation (30) shows that the asymptotic of the deceleration parameter depends only on the entropy parameter $\gamma$ ($b={\pi }^2\gamma /G^2$). We obtain from Equation (29 at $b=0$ ($\gamma =0$) that $q=(3w+1)/2$. The approximate real and positive solutions to Equation (29) for H at $q=0$, $G=1$, $w=-0.1$ are given in Table 1 for some parameters $\gamma$. When $q=0$ the transition redshifts $z_t$, we obtained from Equation (26) at $G=a_0={\rho }_0=1$.

According to Table 1 shows that when the entropy parameter $\gamma$ increases the Hubble parameter H also increases. For a divided point $q=0$ between two pases, universe acceleration and deceleration, the transition reshift $z_t$ is negative and decreases. From Equation (26) we obtain, for the current era when $z=0$, $w=-2/3$, approximate solutions for the Habble parameter H and the deceleration parameter q from Equation (29) for different $\gamma$, presented in Table 2.

In Table 2, negative values of the deceleration parameter q show the acceleration phase of the universe at the current time. The deceleration parameter at the current time is $q_0\approx -0.6$ [36]. In accordance with Table 2 there is entropy parameter $\gamma \approx 0.1$ which can give that result.

5. F(T)-Gravity from Generalized Entropy

Scalar torsion T, in teleparallel gravity, plays a role of a fundamental field similar to the curvature R in Einstein’s general relativity theory [39]. We can describe the inflationary era and the current Universe accelerating expansion by introducing a Lagrangian $F\left(T\right)$ in analogy with $F\left(R\right)$-gravuty. The torsion scalar field T is defined as [40,41]

$$T=S_{\rho }^{\mu \nu }{T}_{\mu \nu }^{\rho },$$

(31)

where tensors $S_{\rho }^{\mu \nu }$ and $T_{\mu \nu }^{\rho }$ are given by

$$S_{\rho }^{\mu \nu }={\displaystyle \frac{1}{2}}\left(K_{\rho }^{\mu \nu }+{\delta }_{\rho }^{\mu }{T}_{\alpha }^{\alpha \nu }-{\delta }_{\rho }^{\nu }{T}_{\alpha }^{\alpha \mu }\right),$$

$$K_{\rho }^{\mu \nu }=-{\displaystyle \frac{1}{2}}\left(T_{\rho }^{\mu \nu }-T_{\rho }^{\nu \mu }-T_{\rho }^{\mu \nu }\right),$$

$$T_{\mu \nu }^{\rho }=e_i^{\rho }\left({\partial }_{\mu }{e}_{\nu }^i-{\partial }_{\nu }{e}_{\mu }^i\right),$$

(32)

where $e_{\nu }^i$ ($i=0,1,2,3$) is a vierbein field, and the metric tensor is $g_{\mu \nu }={\eta }_{ij}{e}_{\mu }^i{e}_{\nu }^j$, where ${\eta }_{ij}$ being the flat metric of the tangent spacetime. For FLRW metric (5), one has $e_{\mu }^i=diag(1,a,a,a)$ anf the torsion scalar field is given by $T=-6H^2$. By variation the action with respect to $e_{\mu }^i$, where the Lagrangian is $F\left(T\right)$, we obtain [42]

$${\displaystyle \frac{1}{6}}\left[F\left(T\right)-2T{F}^{\prime }\left(T\right)\right]{|}_{T=-6H^2}=\left({\displaystyle \frac{8\pi G}{3}}\right)\rho .$$

(33)

Making use of equations (16) and (33) one finds

$$\left[F\left(T\right)-2T{F}^{\prime }\left(T\right)\right]=-T-{\displaystyle \frac{36bT}{36b+T^2}}+12\sqrt{b}arctan\left({\displaystyle \frac{T}{6\sqrt{b}}}\right).$$

(34)

Integrating Equation (34) we obtain the function $F\left(T\right)$:

$$F\left(T\right)=T-{\displaystyle \frac{3\sqrt{3\sqrt{b}T}}{4}}[ln\left({\displaystyle \frac{6\sqrt{b}+T+2\sqrt{3\sqrt{b}T}}{6\sqrt{b}+T-2\sqrt{3\sqrt{b}T}}}\right)$$

$$-2arctan\left(1-\sqrt{{\displaystyle \frac{T}{3\sqrt{b}}}}\right)+2arctan\left(1+\sqrt{{\displaystyle \frac{T}{3\sqrt{b}}}}\right)]$$

$$+12\sqrt{b}arctan\left({\displaystyle \frac{T}{6\sqrt{b}}}\right)+C\sqrt{T},$$

(35)

where C is the constant of integration. As a result, we showed that the teleparallel gravity with the function (35) corresponds to entropic cosmology with entropy (4) proposed.

6. Summary

In conclusion, we have proposed new entropy $S_K=S_{BH}/(1+\gamma S_{BH}^2)$ which possesses similar property as the Bekenstein–Hawking entropy $S_{BH}$; it becomes zero when the apparent horizon radius $R_h$ vanishes. The $S_K$ monotonically increases when the apparent horizon radius $R_h$ increases and $S_K$ is positive. We have studied the barotropic perfect fluid with flat FLRW universe. By exploring the first law of apparent horizon thermodynamics we obtained the modified Friedmann’s equations. We have the addition term in the second Friedmann’s equation which is a dynamical cosmological constant. We have showed that holographic dark energy is the source of the universe inflation. It is worth mentioning that Barrow’s and Tsallis’s entropies also lead to Einsten’s equations with the dynamical cosmological constant [35]. We have found that for some parameters our model have phases of universe inflation and deceleration and eternal inflation. The transition redshifts when $q=0$, presented in Table 1 were calculated for some EoS parameter w and for entropy parameter $\gamma$. According to Table 2, at $\gamma \approx 0.1$ and $w=-2/3$ the current deceleration parameter $q_0\approx -0.6$ is realised. It was shown that dynamical cosmological constant gives EoS of dark energy $w_D=-1$ at large Habble parameter H (small apparent horizon radius $R_h$). So, after Big Bang the de Sitter space takes place ($p_D+{\rho }_D=0$) and universe inflation is due to dynamical cosmological constant. We have showed that when $R_h\to \infty$ ($H\to 0$) dynamical cosmological constant vanishes (${\Lambda }_{eff}\to 0$). It is shown that entropic cosmology with our entropy proposed is equivalent to cosmology based on the teleparallel gravity with the function $F\left(T\right)$. It is worth noting that similar results were discussed in other models [37,38]. Thus, cosmology based on the modified Friedmann equations obtained may be of interest for a description of inflation and late time universe acceleration.